Calculus Examples

$\int_{0}^{1} x \sin (x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sin (x)$ .

$x (- \cos (x))]_{0}^{1} - \int_{0}^{1} - \cos (x) d x$

Step 2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$x (- \cos (x))]_{0}^{1} - - \int_{0}^{1} \cos (x) d x$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Multiply $- 1$ by $- 1$ .

$x (- \cos (x))]_{0}^{1} + 1 \int_{0}^{1} \cos (x) d x$

Step 3.2

Multiply $\int_{0}^{1} \cos (x) d x$ by $1$ .

$x (- \cos (x))]_{0}^{1} + \int_{0}^{1} \cos (x) d x$

Step 4

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

$x (- \cos (x))]_{0}^{1} + \sin (x)]_{0}^{1}$

Step 5

Simplify the answer.

Tap for more steps...

Step 5.1

Combine $x (- \cos (x))]_{0}^{1}$ and $\sin (x)]_{0}^{1}$ .

$x (- \cos (x)) + \sin (x)]_{0}^{1}$

Step 5.2

Substitute and simplify.

Tap for more steps...

Step 5.2.1

Evaluate $x (- \cos (x)) + \sin (x)$ at $1$ and at $0$ .

$(1 (- \cos (1)) + \sin (1)) - (0 (- \cos (0)) + \sin (0))$

Step 5.2.2

Simplify.

Tap for more steps...

Step 5.2.2.1

Multiply $- 1$ by $1$ .

$- \cos (1) + \sin (1) - (0 (- \cos (0)) + \sin (0))$

Step 5.2.2.2

Multiply $- 1$ by $0$ .

$- \cos (1) + \sin (1) - (0 \cos (0) + \sin (0))$

Step 5.2.2.3

Multiply $0$ by $\cos (0)$ .

$- \cos (1) + \sin (1) - (0 + \sin (0))$

Step 5.2.2.4

Add $0$ and $\sin (0)$ .

$- \cos (1) + \sin (1) - \sin (0)$

Step 5.3

Simplify.

Tap for more steps...

Step 5.3.1

The exact value of $\sin (0)$ is $0$ .

$- \cos (1) + \sin (1) - 0$

Step 5.3.2

Multiply $- 1$ by $0$ .

$- \cos (1) + \sin (1) + 0$

Step 5.3.3

Add $- \cos (1) + \sin (1)$ and $0$ .

$- \cos (1) + \sin (1)$

Step 5.4

Simplify.

Tap for more steps...

Step 5.4.1

Evaluate $\cos (1)$ .

$- 1 \cdot 0.5403023 + \sin (1)$

Step 5.4.2

Multiply $- 1$ by $0.5403023$ .

$- 0.5403023 + \sin (1)$

Step 5.4.3

Add $- 0.5403023$ and $\sin (1)$ .

$0.30116867$